Summer Semester 2022

Notes

1. Outline and Bayesian inversion example
2. Gaussians
3. Probability metrics
4. Random dynamical systems I
5. Random dynamical systems II
6. Smoothing problem
7. Filtering problem
8. Linear systems I: Kalman filter
9. Asymptotics for the (scalar) Kalman filter
10. Ill-posedness of (some) inverse problems
11. Well-posedness of smoothing and filtering
12. MAP estimator and posterior mean
13. Linear systems II: Kalman smoother
14. Markov Chain Monte Carlo
15. Metropolis-Hastings algorithm
16. Metropolis-adjusted Langevin algorithm (MALA)
17. Independence Sampler and precoditioned Crank-Nicolson scheme
18. Example: logistic map
19. Variational methods
20. Numerical illustration of smoothing algorithms
21. Approximate Gaussian filters
22. Particle filters
23. Convergence of the bootstrap filter
24. Variants of the bootstrap filter

Matlab files

• Example #1: linear 2D system
• Example #2: sine map
• Example #3: Lorenz' 63 system
• Example #4: smoothing (deterministic in 1D)
• Example #5: Kalman filter in 2D
• Example #6: linear 1D system
• Example #7: logistic map (smoothing posterior and RWM)
• Example #8: Independence Sampler for the sine map
• Example #9: preconditioned Crank-Nicolson scheme for sine map
• Example #10: 4DVAR for the sine map
• Example #11: particle filter(s) for the sine map

Lecture notes (2019)

Carsten Hartmann, Lorenz Richter. Uncertainty Quantification, 2019 (online).

Projects

• Hamiltonian Monte Carlo (e.g. Neal, MCMC Using Hamiltonian Dynamics, 2011)
• MCMC for functions (e.g. Cotter et al, MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster, 2013)
• Linear models and regularisation (e.g. Chapter 6.1 in: Sullivan, Introduction to Uncertainty Quantification, 2015)
• Ensemble Kalman Filter (e.g. Katzfuss et al, Understanding the Ensemble Kalman Filter, 2016)
• MAP estimator vs. Bayes estimator (e.g. Bassett & Deride, Maximum a Posteriori Estimators as a Limit of Bayes Estimators, 2019)
• Continuous-time filtering (e.g. Chapter 6.4 in: Law et al, Data Assimilation: A Mathematical Introduction, 2016)

Further reading

• Tim J. Sullivan, Introduction to Uncertainty Quantification, Springer, 2015 (online).
• Sebastian Reich, Colin Cotter, Probabilistic Forecasting and Bayesian Data Assimilation, Cambridge University Press, 2015.
• Kody Law, Andrew Stuart, Kostas Zygalakis, Data Assimilation: A Mathematical Introduction, Springer, 2016 (online).